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Binomial Coefficient

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For $ n,k \in{\mathbb{N}}_0$ with $ n \geq k$ the binomial coefficient $ \binom{n}{k} $ is defined as

$\displaystyle \binom{n}{k} = \frac{n!}{(n-k)!k!} = \frac{n(n-1)(n-2)\cdots(n-k+1)}{1 \cdots (k-2)(k-1)k}\,. $

With $ 0! = 1$ we have in particular

$\displaystyle \left(\begin{array}{c}0 \\ 0\end{array} \right) = 1, \quad \left... ...\ n\end{array} \right) = \left(\begin{array}{c}n \\ 0\end{array}\right) = 1 . $

The binomial coefficient $ \binom{n}{k} $ equals the number of $ k$-subsets of a set containing $ n$ elements.

(Authors: Kimmerle/Abele)
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  automatically generated 6/11/2007